# Gravity affecting gravity?

After reading, I'm actually still not sure what it is this thread is about But, it sounds like perhaps you are interested in the Weyl Tensor. It describes the curvature due intrinsically to the gravitational field. Here is the Wiki article for your convenience.

http://en.wikipedia.org/wiki/Weyl_tensor" [Broken]

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Guys, the reason why you are arguing is because gravitational energy can't be localized. You can only talk about the total energy of the system. You can't split it up into "matter energy" and "field energy" (or, put another way, there are too many inequivalent ways to try to do this--hence all your arguing). This is a standard topic covered in GR books. The most challenging part about learning GR is learning the right questions to ask.

That said, the physical intuition in the OP is sound. If you view the theory perturbatively off of flat spacetime then there is a sense in which this "iterative" procedure described in the OP does take place. In fact there is a famous derivation (really more of a plausibility argument) due to Feynman whereby you begin with a linear equation and add "gravity gravitates" to boostrap your way up to the full nonlinear Einstein equations.

If you view the theory perturbatively off of flat spacetime then there is a sense in which this "iterative" procedure described in the OP does take place. In fact there is a famous derivation (really more of a plausibility argument) due to Feynman whereby you begin with a linear equation and add "gravity gravitates" to boostrap your way up to the full nonlinear Einstein equations.

There we go! That's exactly what I wanted to know. What are these equations called? (I don't know anything about general relativity -- is it the same thing?)

Guys, the reason why you are arguing is because gravitational energy can't be localized. You can only talk about the total energy of the system. You can't split it up into "matter energy" and "field energy" (or, put another way, there are too many inequivalent ways to try to do this--hence all your arguing). This is a standard topic covered in GR books. The most challenging part about learning GR is learning the right questions to ask.

I don't think this is true... The Ricci tensor, or equivalently the stress-energy-momentum tensor, is precisely the object which describes the energy not due to the gravitational field, However, a separate object exists which does include that energy.

See here http://en.wikipedia.org/wiki/Stress-energy-momentum_pseudotensor" [Broken]

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There we go! That's exactly what I wanted to know. What are these equations called? (I don't know anything about general relativity -- is it the same thing?)

if you want to ask gravity, and relativity questions, you should study GR, since this is exactly what it covers.

pervect
Staff Emeritus
I don't think this is true... The Ricci tensor, or equivalently the stress-energy-momentum tensor, is precisely the object which describes the energy not due to the gravitational field, However, a separate object exists which does include that energy.

See here http://en.wikipedia.org/wiki/Stress-energy-momentum_pseudotensor" [Broken]

Because it's not a true tensor, the stress-energy tensor isn't coordinate independent (and IIRC it even depends on the gauge). In any event, it won't give you a unique answer for the distribution of energy, because it's not a true tensor and hence not coordinate independent.

So Sam's answer is spot-on, and very well written.

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if you want to ask gravity, and relativity questions, you should study GR, since this is exactly what it covers.

Well, I didn't realize it was a GR question. My post just got moved to this forum.

Because it's not a true tensor, the stress-energy tensor isn't coordinate independent (and IIRC it even depends on the gauge). In any event, it won't give you a unique answer for the distribution of energy, because it's not a true tensor and hence not coordinate independent.

So Sam's answer is spot-on, and very well written.

I'll have to do some looking into that, I'm sorry to doubt, but wiki gives an explanation as to why what you said isn't true, and I need more time because of my naivete to be sure.

Thanks all the same.

EDIT: also, I guess I'm a little confused by your post since you segued with "because". Are you saying that the stress-energy-psudeo-tensor does reflect an object that represents the energy of the gravitational field. It seems like you wrote "the SEM-pseudotensor is that object 'because'..." but your post seems to object to what I said. Could you clarify what you meant up above.

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Perhaps you are talking about gravitational waves. Gravitational waves carry energy just like electromagnetic waves, and it is gravitating.

zonde
Gold Member
Guys, the reason why you are arguing is because gravitational energy can't be localized.
And why do you think that there is any "gravitational energy" at all?

You can only talk about the total energy of the system. You can't split it up into "matter energy" and "field energy" (or, put another way, there are too many inequivalent ways to try to do this--hence all your arguing).
If you can attribute all of the total energy of the system to "matter energy" plus radiation energy without anything left then there is no reason to think that there anything like "field energy".

And why do you think that there is any "gravitational energy" at all?

If you can attribute all of the total energy of the system to "matter energy" plus radiation energy without anything left then there is no reason to think that there anything like "field energy".

There is such a thing as "gravitational energy" because gravitational waves carry it. You can precisely characterize the rate at which an isolated system is losing energy to gravitational-wave emission. What you can't do is say that so much of the energy was at one place in spacetime and so much of the energy was elsewhere. or whether the energy "came from" matter or field energy. (Of course, you could try to make such a statement using a "psuedotensor" as somebody brought up, but somebody else could come along with another inequivalent pseudotensor and claim that *that* one was the true gravitational energy. What everyone will agree about is the total energy in the system as well as its rate of change.) People do use "stress-energy pseudotensors" in computations, but that's mainly for convenience in performing some specific computation. All claims about a "true" local gravitational energy density have disappeared from the literature at this point. (There is a community still looking for "quasi-local" gravitational energy, but that doesn't seem to work very well either.)

You may be trying to call this sort of energy "radiation energy" so you don't need "field energy". But there are reasons for saying that even a non-radiating system has "gravitational binding energy" (that can't be localized). For example, in the Newtonian limit of GR the energy/mass of an isolated system can be written as an integral over the mass density of the matter. But as soon as you go back to GR (or just a post-Newtonian correction), this property is lost. What is the other contribution? It makes sense to think of it as energy in the field. But there's no way to write it as an integral over matter plus an integral over field, so again it makes sense to say "gravitational energy exists but can't be localized".

pervect
Staff Emeritus
Do you happen to have MTW's textbook, "Graitation"? I know they have a good section on the topic of why you can't localize the energy of a gravitational field.

I'm not sure how much more clear I can be, but I'll try saying it again.

Covariance is an important physical principle. It boils down to saying that measurements made by different observers represnt the same underlying reality.

The sort of covariance we need for relativity is Lorentz covariance. Any four-vector, regardless of whether it is (time, distance) or (energy, momentum) must transform via the lorentz transforms to have a physical meaning that's independent of the coordinate system.

If you don't have covariance, your quantity cannot be defined in an observer independent way.

Pseudotensors, in the sense used in General Relatiavity (i.e. the energy pseudotensor you refer to) do not define energy in a way that's independent of the observer.

Some people have remarked, with some merit, that the "pseudotensors" in GR are really just non-tensors.

The thing that makes energy pseudotensors useful at all is that while they don't offer an observer-compatible defintion of energy, the total energy computed via them will transform properly given the proper conditions (usually asymptotic flatness).

So the pseudotensors themselves do not offer any physically meaningful way to localize energy because different observers will, as other posters have remarked, not have compatible views of how the energy is distributed.

They do allow you to come up with a total energy that everyone agrees on, however. (And they aren't the only method of doing it, there's a proof I think that the pseudotensor definition of energy matches the Bondi defiition).

zonde
Gold Member
There is such a thing as "gravitational energy" because gravitational waves carry it. You can precisely characterize the rate at which an isolated system is losing energy to gravitational-wave emission.
Isolated system can't loose energy.
Actually we don't have truly isolated systems so in order to model such a system we can imagine system in environment that mirrors all the outward interactions of the system with equivalent inward interactions.
This means that if the system is emitting gravitational waves then equivalent waves are directed toward the system from environment. And I would say that if gravitating system can emit these waves then it can absorb the same waves unless you have some strong arguments why this shouldn't be so.

What you can't do is say that so much of the energy was at one place in spacetime and so much of the energy was elsewhere. or whether the energy "came from" matter or field energy. (Of course, you could try to make such a statement using a "psuedotensor" as somebody brought up, but somebody else could come along with another inequivalent pseudotensor and claim that *that* one was the true gravitational energy. What everyone will agree about is the total energy in the system as well as its rate of change.) People do use "stress-energy pseudotensors" in computations, but that's mainly for convenience in performing some specific computation. All claims about a "true" local gravitational energy density have disappeared from the literature at this point. (There is a community still looking for "quasi-local" gravitational energy, but that doesn't seem to work very well either.)
What I say is that mass is moving down in gravitational potential when the system emits gravitational waves. So it's mass that is loosing this energy. The same way if you would try to reverse this process it's mass where you have to put this energy to move it upwards in gravitational potential.

You may be trying to call this sort of energy "radiation energy" so you don't need "field energy". But there are reasons for saying that even a non-radiating system has "gravitational binding energy" (that can't be localized). For example, in the Newtonian limit of GR the energy/mass of an isolated system can be written as an integral over the mass density of the matter. But as soon as you go back to GR (or just a post-Newtonian correction), this property is lost. What is the other contribution? It makes sense to think of it as energy in the field. But there's no way to write it as an integral over matter plus an integral over field, so again it makes sense to say "gravitational energy exists but can't be localized".
I have problems with that. "Gravitational binding energy" is negative energy. So we have missing energy not excess energy.

Thanks pervect,

your helpful as always. I do have that book and I take a look at that section.